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We investigate when a complete graph $K_n$ with some edges deleted is determined by its adjacency spectrum. It is shown to be the case if the deleted edges form a matching, a complete graph $K_m$ provided $m \leq n-2$, or a complete…

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The…

We prove that any graph $G$ of minimum degree greater than $2k^2-1$ has a $(k+1)$-connected induced subgraph $H$ such that the number of vertices of $H$ that have neighbors outside of $H$ is at most $2k^2-1$. This generalizes a classical…

We introduce a hypergraph matrix, named the unified matrix, and use it to represent the hypergraph as a graph. We show that the unified matrix of a hypergraph is identical to the adjacency matrix of the associated graph. This enables us to…

A graph $G$ is $\{F_{1}, F_{2},\dots,F_{k}\}$-free if $G$ contains no induced subgraph isomorphic to any $F_{i}$ $(1\leq i \leq k)$. A connected graph $G$ is a split graph if its vertex set can be partitioned into a clique and an…

Let $G$ be a connected simple graph on $n$ vertices and $m$ edges. Denote $N_{i}^{(j)}(G)$ the number of spanning subgraphs of $G$ having precisely $i$ edges and not more than $j$ connected components. The graph $G$ is \emph{strong} if…

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…

A graph $\Gamma$ is $k$-connected-homogeneous ($k$-CH) if $k$ is a positive integer and any isomorphism between connected induced subgraphs of order at most $k$ extends to an automorphism of $\Gamma$, and connected-homogeneous (CH) if this…

A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the…

A relational structure is (connected-)homogeneous if every isomorphism between finite (connected) substructures extends to an automorphism of the structure. We investigate notions which generalise (connected-)homogeneity, where…

A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$ than $v$. The boundary $\partial(G)$ of $G$ is the set of all of its boundary…

The adjacency-diametrical matrix (AD matrix) of a connected graph $G$ with diameter $d$, denoted by $AD(G)$, is the matrix indexed by the vertices of $G$ in which the $(i,j)$-entry of $AD(G)$ is $1$ if $d_G(v_i,v_j)=1$, is $d$ if…

Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$…

The difference between the two largest eigenvalues of the adjacency matrix of a graph $G$ is called the spectral gap of $G.$ If $G$ is a regular graph, then its spectral gap is equal to algebraic connectivity. Abdi, Ghorbani and Imrich, in…

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of…

We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\geq |E(H)|^2$. We use this to prove that every three-coloring of the edges of a complete graph…

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix.…

Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and…

For a graph $G$ and a set of graphs $\mathcal{H}$, we say that $G$ is {\em $\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\mathcal{F}$,…

For a graph $G$, let $f(G)$ be the largest integer $k$ for which there exist two vertex-disjoint induced subgraphs of $G$ each on $k$ vertices, both inducing the same number of edges. We prove that $f(G) \ge n/2 - o(n)$ for every graph $G$…